The angles of quadrilateral $ABCD$ satisfy  $\angle A = 2\angle B =
3\angle C = 4\angle D$.  What is the degree measure of $\angle A$, rounded to the nearest whole number?
Answer: Let $x$ be the degree measure of $\angle A$.  Then the degree measures of angles $B$, $C$,  and $D$ are $x/2$, $x/3$, and $x/4$, respectively.  The degree measures of the four angles have a sum of 360, so \[
360 = x+\frac{x}{2}+\frac{x}{3}+\frac{x}{4} =
\frac{25x}{12}.
\]Thus $x=(12\cdot 360)/25 = 172.8\approx \boxed{173}$.